Scaling Properties of Delay Tolerant Networks with Correlated Motion Patterns Uichin Lee, Bell Labs/Alcatel-Lucent Soon Y. Oh, Mario Gerla (UCLA) Kang-Won Lee (IBM T.J. Watson Research)

Alcatel-Lucent 2 | CHANTS’09 | Sept Key DTN Metric: Pair-wise Inter-contact Time Key metric for measuring end-to-end delay Pair-wise inter-contact time: interval between two contact points Inter-contact distribution:  Exponential : bounded delay, but not realistic?  Power-law  : may cause infinite delay, but more realistic? T1 T2 T3 Contact points between two nodes: i and j T ic (n): pair-wise inter-contact time time T ic (1) T1 T ic (2) T2 T ic (3) T3

Alcatel-Lucent 3 | CHANTS’09 | Sept Two-phase Inter-contact Time Two-phase distribution: power-law head and exponential tail Levy walk based mobility Lee et al. Infocom’08/09 Inter-contact time CCDF Time (min) Power Law Head Exponential Tail Power Law Head Inter-contact time CCDF Exponential Tail Karagiannis et al., MobiCom’07 Association times w/ AP (UCSD) or cell tower (MIT cell) Direct contact traces: Infocom, cambridge (imotes), MIT-bt Power Law Head Log-log plot Time (days) Log-log plot

Alcatel-Lucent 4 | CHANTS’09 | Sept Two-phase Inter-contact Time Why two-phase distribution? One possible cause:  Flight distance of each random trip (within a finite area) [Cai08]  The shorter the flight distance, the higher the motion correlation in local area, resulting heavier power-law head  Power-law head while in local area vs. exponential tail for future encounters Examples of motion correlation:  Manhattan sightseers: In Time Square, sightseers tend to bump into each other; and then depart for other sights  Levy flight of human walks [Rhee08]: short flights + occasional long flights  Vehicular mobility: constrained by road traffic (+traffic jam)  High correlation among vehicles in close proximity  After leaving locality, vehicles meet like “ ships in the night ” *Cai08: Han Cai and Do Young Eun, Toward Stochastic Anatomy of Inter-meeting Time Distribution under General Mobility Models, MobiHoc’08 *Rhee08: Injong Rhee, Minsu Shin, Seongik Hong, Kyunghan Lee and Song Chong, On the Levy-walk Nature of Human Mobility, INFOCOM’08 Goal: understanding impacts of correlated motion patterns on throughput/delay/buffer scaling properties Frequent encounters in local area Local area

Alcatel-Lucent 5 | CHANTS’09 | Sept DTN Model: Inter-contact rate + flight distance Inter-contact rate: λ ~ speed (v)*radio range (r) [Groenevelt, Perf’05]  Random waypoint/direction (exp inter-contact time) Flight distance (motion correlation):  Ranging from radio radius Ω(r) to network width O(1) Invariance property: avg inter-contact time does not depend on the degree of correlation in the mobility patterns [Cai and Eun, Mobihoc’08] 1 1 flight distance = Θ(1) Random direction Random walk 1 1 flight distance = Θ(r)radio range=Θ(r) Varying flight distance

Alcatel-Lucent 6 | CHANTS’09 | Sept Hop Relay: DTN Routing Each source has a random destination (n source-destination pairs) 2-hop relay protocol: 1. Source sends a packet to a relay node 2. Relay node delivers a packet to the corresponding receiver 2-hop Relay by Grossglauser and Tse Source Relay Destination Source to Relay Relay to Dest

Alcatel-Lucent 7 | CHANTS’09 | Sept Hop Relay: Throughput Analysis Intuition: avg throughput is determined by aggregate meeting rate (src  relay and relay  dest) Two-hop relay per node throughput : Θ(nλ)  Aggregate meeting rate at a destination: nλ  Grossglauser and Tse’s results: Θ(nλ)=Θ(1) when λ = 1/n (i.e., speed 1/√n, radio range 1/√n) Motion correlation (w/ flight len [O(r),O(1)]) still preserves meeting rate Source to RelayRelay to Destination Source Destination Food (=source) Ants (=relays)

Alcatel-Lucent 8 | CHANTS’09 | Sept Hop Relay: Delay Analysis Source to relay node (D sr ), and then relay to dest (D rd )  D sr = avg. inter-any-contact time to a random relay  D rd = avg. residual inter-contact time (relay  dest)  Mean residual inter-contact time (D rd ) = E[T 2 ] / 2E[T] >> T is a random variable for inter-contact time  Random Direction (RD): E[T 2 ] = O(1/λ 2 )  D rd =O(1/λ)  Random Walk (RW): E[T 2 ] = O(logn*1/λ 2 )  D rd =O(logn/λ) Average 2-Hop Delay: D(n) = [O(1/λ), O(logn/λ)]  As flight distance increases, average delay decreases monotonically Inspection paradox (length bias): source tends to sample a longer inter-contact interval between relay and dest nodes Time Src  Relay Inter-contact history: Relay  Dest Source Relay Destination Source to Relay Relay to Dest

Alcatel-Lucent 9 | CHANTS’09 | Sept Hop Relay: Buffer Requirement Little’s law: buffer = (rate) x (delay) Required buffer space per node: B = [Θ(n), Θ(nlogn)]  Rate*delay = Θ(nλ)* [1/λ, logn/λ] = [Θ(n), Θ(nlogn)]  Recall: random direction (1/λ) and random walk (logn/λ)  Motion correlation requires more buffer space  Correlation increases burstiness of relay traffic Impact of limited buffer space on throughput  Limited buffer of Θ(K) at each relay node is fairly shared by all sources  Relay node carries a packet to a certain dest with prob. K/B  Throughput per source = Θ(nλ*K/B) Delay: [Θ(1/λ), Θ(logn/λ)] S1 S2 R Θ(n) srcs Relay S3 D1 D2 D3 Θ(n) dests Rate: Θ(nλ)

Alcatel-Lucent 10 | CHANTS’09 | Sept Simulation: Throughput Degree of correlation via average flight distance L  L=250m  high correlation  power law head + exponential tail  L=1000m  low correlation  almost exponential Throughput is independent of the degree of correlations Average throughput per node as a function of # nodes Throughput per node (kbps) 5000m*5000m area # of nodes <=100 Radio range = 250m w/ 2Mbps CCDF of inter-contact time (20m/s) L=250m log-linear L=1000m L=250m Exp Inter-contact time CCDF Log-log

Alcatel-Lucent 11 | CHANTS’09 | Sept Simulation: Inter-any-contact Time Inter-any(k)-contact time: inter-contact time to any of k nodes Invariance property: avg inter-contact time is independent of correlation Residual inter-contact time: source probes a random point of the inter- meeting times between relay and destination Average Inter-contact time Average residual inter-contact time 20m/s 30m/s 20m/s L=250m 20m/s L=1000m 30m/s L=250m 30m/s L=1000m Number of relay nodes (k)

Alcatel-Lucent 12 | CHANTS’09 | Sept Simulation: Buffer Utilization Burstiness of relay traffic increases with the degree of correlation SSDDDSSD N=3 N=2 Relay node’s contact history Time # consecutive encounters (N) S Source D Destination N=2 Cumulative dist of # consecutive encounters L=250 L=1000 Number of consecutive encounters (N) P(N<=X)

Alcatel-Lucent 13 | CHANTS’09 | Sept Conclusion Impact of correlated motion patterns on DTN scaling properties DTN model: inter-contact rate + motion correlation via flight distance  Flight distance of Ω(r); i.e., min travel distance ~ one ’ s radio range  Considered mobility ranges from Random Walk to Random Direction Main results:  Throughput is independent of motion correlation  Delay monotonically increases with the degree of correlation  Buffer requirement also increases with the degree of correlation  Correlation increases burstiness of relay traffic Future work:  Applying results to DTN multicast scenarios  Scaling properties of inter-domain DTN scenarios